On the Analytic Continuation of Appell’s Hypergeometric Function <i>F</i>₂ to Some Symmetric Domains in the Space <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mn>2</mn></msup></semantics></math></inline-formula>

Abstract

The paper considers the problem of representation and extension of Appell’s hypergeometric functions by a special family of functions—branched continued fractions. Here, we establish new symmetric domains of the analytical continuation of Appell’s hypergeometric function F2 with real and complex parameters, using their branched continued fraction expansions whose elements are polynomials in the space C2. To do this, we used a technique that extends the domain of convergence of the branched continued fraction, which is already known for a small domain, to a larger domain, as well as the PC method to prove that it is also the domain of analytical continuation. A few examples are provided at the end to illustrate this.

Keywords

• Appell’s hypergeometric function
• branched continued fraction
• analytic continuation
• convergence
• approximation by rational functions